Problem 27 - Precalculus with Calculus Previews

Question

Suppose that \(1 \&\) is deposited in a savings account paying \(1 \%\) annual interest compounded continuously. How much money will have accrued in the account after 2000 years? What is the future value of \(1 \&\) in 2000 years if the account pays \(2 \%\) annual interest compounded continuously?

Step-by-Step Solution

Answer

Future value after 2000 years: \( 485165195.41 \) units at 1% interest; approximately \( 2.35\times10^{17} \) units at 2% interest.

Step 1: Understand Continuous Compounding Formula

The formula used to calculate compound interest with continuous compounding is given by \( A = Pe^{rt} \), where \( A \) stands for the amount of money accumulated after n years, including interest. \( P \) is the principal amount (the initial deposit), \( r \) is the annual interest rate (in decimal), and \( t \) is the time the money is invested for in years.

Step 2: Insert Values for 1% Interest Rate

In this case, the principal \( P = 1 \) unit, the rate \( r = 0.01 \), and the time \( t = 2000 \) years. Plug these into the continuous compounding formula: \( A = 1 \cdot e^{0.01 \times 2000} = e^{20} \).

Step 3: Calculate \( e^{20} \)

Using a calculator, compute \( e^{20} \) to determine the future value at a 1% interest rate compounded continuously. This value is approximately 485165195.41 units.

Step 4: Insert Values for 2% Interest Rate

Now, calculate for an interest rate of 2%. Here, \( P = 1 \), \( r = 0.02 \), and \( t = 2000 \). Use the formula: \( A = 1 \cdot e^{0.02 \times 2000} = e^{40} \).

Step 5: Calculate \( e^{40} \)

Use a calculator to find \( e^{40} \), which is the future value at a 2% interest rate compounded continuously. This value is approximately 2.353852668×10^{17} units.

Key Concepts

Compound Interest Formula

The compound interest formula is crucial in finance for calculating the growth of an investment over time. In traditional compounding, interest is added at specific intervals, such as yearly or monthly. However, with continuous compounding, interest is added at every possible instant. This is represented using the formula:

• \( A = Pe^{rt} \)

Here, \( A \) is the amount of money accumulated after a certain time, \( P \) is the principal or initial amount of money, \( r \) represents the annual interest rate expressed as a decimal, and \( t \) is the time for which the money is invested, expressed in years.
This formula assumes that the interest compound continuously, which essentially means the account is always accruing interest. This provides a more precise method for interest accumulation, particularly useful for high-frequency financial strategies.
Understanding this concept is critical for calculating how much an investment grows over time with continuous interest, as it forms the basis for accurate future value calculations in the context of finance.

Future Value Calculation

Future value calculation allows us to determine how much a current investment will grow over a specific period of time. With continuous compounding, this is particularly effective as it accounts for the highest possible compounding frequency.
To calculate future value using continuous compounding, one inserts values into the compound interest formula:

• For an interest rate of 1%, substituting \( P = 1 \), \( r = 0.01 \), and \( t = 2000 \) into the formula gives us \( A = 1 \cdot e^{0.01 \times 2000} = e^{20} \).
• Using a similar method for a 2% interest rate, \( A = 1 \cdot e^{0.02 \times 2000} = e^{40} \).

The calculation of \( e^{20} \) gives a future value of approximately 485165195.41 units, and \( e^{40} \) results in an approximate future value of 2.353852668\( \times 10^{17} \) units. These calculations emphasize the dramatic effect of compounding, especially when projected over a long duration such as 2000 years.

Exponentiation in Finance

Exponentiation is a mathematical operation involving a base and an exponent. In finance, exponentiation is a key component of the compound interest formula used for continuous compounding. The exponential function \( e^{rt} \) represents the natural growth of funds over time.
When interest compounds continuously, the growth of the investment is modeled exponentially. This is because the formula uses Euler's number \( e \), which approximates to 2.71828. It signifies the base rate of growth shared by all continually growing processes.
This is why both \( e^{20} \) and \( e^{40} \) are critical in determining future values in our exercise. They denote the scale of growth for 1% and 2% interest rates over 2000 years. Exponentiation effectively highlights how even small differences in the interest rate or compounding rate can lead to vastly different outcomes.